Problem 71 - University Physics with Modern Physics

Question

In. A vacuum cleaner belt is looped over a shaft of radius 0.45 \(\mathrm{cm}\) and a wheel of radius 2.00 \(\mathrm{cm} .\) The arrangement of the bell, shaft, and wheel is similar to that of the chain and sprockets in Fig. 9.14 . The motor turns the shaft at 60.0 rev/s and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed. Assume that the belt doesn't ship on either the shaft or the wheel. (a) What is the speed of a point on the belt? (b) What is the angular velocity of the wheel, in rad/s?

Step-by-Step Solution

Answer

(a) 54π cm/s; (b) 27π rad/s.

Step 1: Calculate the Linear Speed of the Belt

Since the belt does not slip, the linear speed of the belt is equal to the linear speed at the edge of the shaft. First, we need to calculate the circumference of the shaft with radius 0.45 cm, which is given by the formula:\[ \text{Circumference of shaft} = 2\pi r_\text{shaft} = 2\pi \times 0.45 \, \text{cm} \].Then, the linear speed \(v\) is calculated using the revolution per second:\[ v = \text{revolutions per second} \times \text{circumference} = 60.0 \, \text{rev/s} \times 2\pi \times 0.45 \, \text{cm} \].Solving this gives:\[ v = 60.0 \times 2\pi \times 0.45 \, \text{cm/s} = 54\pi \, \text{cm/s}\].

Step 2: Calculate the Angular Velocity of the Wheel

The linear speed of the belt is equal to the linear speed around the wheel's circumference since the belt doesn't slip. Thus, the linear speed \(v\) can also be expressed in terms of the wheel's radius \(r_\text{wheel}\) and its angular velocity \(\omega\):\[ v = r_\text{wheel} \times \omega\text{.} \]We have already found \(v = 54\pi \, \text{cm/s}\). The radius of the wheel \(r_\text{wheel}\) is \(2.00\, \text{cm}\). Thus, \(\omega\) is given by:\[ \omega = \frac{v}{r_\text{wheel}} = \frac{54\pi}{2.00} \, \text{rad/s} \].Simplifying, we find:\[ \omega = 27\pi \, \text{rad/s}\].

Key Concepts

Angular Velocity

Angular velocity is a measure of how fast an object rotates or revolves. It tells us the speed of rotation in terms of angle per unit time.
Unlike linear velocity, which measures the rate of change of position along a straight path, angular velocity is concerned with rotational motion.

• Units: Typically measured in radians per second (rad/s).
• Formula: If you know the linear speed (v) of a point on the rotating object and the radius (r), you can find angular velocity (\( \omega \)) using the formula: \[ \omega = \frac{v}{r} \]

In our vacuum cleaner scenario, the motor spins a shaft, and ultimately wheels and rollers, at a certain speed. Knowing the shaft and wheel dimensions helps us calculate their angular velocities.
This tells us how fast the wheel rotates when driven by the motor.

Linear Speed

Linear speed refers to the rate of movement along a path or a straight line.
In rotational systems like gears or belts, it is the speed at which a point on the outer edge moves.

• Relation to Rotation: Linear speed comes into play when considering wheel perimeters or belt lengths, helping to bridge linear and rotational motion.
• Formula: For an object moving in a circle, its linear speed (\( v \)) can be found using: \[ v = \text{circumference} \times \text{revolutions per second} \]

In the context of the vacuum cleaner, the shaft and belt share the same linear speed since there is no belt slipping.
Knowing the shaft's parameters allows for the calculation of the belt's speed, which is essential in determining other motion aspects in the system.

Rotational Motion

Rotational motion happens when an object spins around an internal or external axis.
Unlike translational motion, which involves object shifting linearly, rotational motion is about circular movement.

• Components: Important aspects include angular velocity, torque, and rotational inertia.
• Application: Pulleys, gears, and wheels all exhibit rotational motion. The dynamics of objects like these are determined by rotational parameters.

Our problem involves rotational motion via the vacuum cleaner's belt system.
This motion drives the wheel and, in turn, enables the vacuum's roller to perform its function effectively.
Understanding these rotational principles aids in grasping machine operations from household items to complex machinery systems.